Consider the sequence,

MATHEMATICS · Subjective · A Level · Mathematics · Sequences and series · Sequences and series

Problem

Consider the sequence,

\boldsymbol{b}_{\boldsymbol{n}}

, defined recursively as follows:

b_{1}=3

,

b_{2}=2

, and

b_{n}=\frac{b_{n-1}}{b_{n-2}}

. a. Complete the table below: <table className='table-auto pt-3 border-collapse border w-full'> <tr> <th className='border px-4 py-2 bg-[#f3f4f6] text-left'>

b_{1}

</th> <th className='border px-4 py-2 bg-[#f3f4f6] text-left'>

b_{2}

</th> <th className='border px-4 py-2 bg-[#f3f4f6] text-left'>

b_{3}

</th> <th className='border px-4 py-2 bg-[#f3f4f6] text-left'>

b_{4}

</th> <th className='border px-4 py-2 bg-[#f3f4f6] text-left'>

b_{5}

</th> <th className='border px-4 py-2 bg-[#f3f4f6] text-left'>

b_{6}

</th> <th className='border px-4 py-2 bg-[#f3f4f6] text-left'>

b_{7}

</th> <th className='border px-4 py-2 bg-[#f3f4f6] text-left'>

b_{8}

</th> <th className='border px-4 py-2 bg-[#f3f4f6] text-left'>

b_{9}

</th> </tr> <tr> <td className='border px-4 py-2'>3</td> <td className='border px-4 py-2'>2</td> <td className='border px-4 py-2'></td> <td className='border px-4 py-2'></td> <td className='border px-4 py-2'></td> <td className='border px-4 py-2'>

\dfrac{3}{2}

</td> <td className='border px-4 py-2'></td> <td className='border px-4 py-2'></td> <td className='border px-4 py-2'></td> </tr> </table> b. Using your work from part (a), evaluate the series below:

\sum_{n=7}^{73} b_{n}

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